If only one of the distributions \(L,\tilde{L}\)  is integrable,
it is worth mentioning that, apart from the above interpretation in terms of the twistor programme,
(complex) 4-manifolds admitting a foliation by the associated twistor surfaces have been extensively studied by
Plebański and collaborators, who called these surfaces null strings,
and the manifolds admitting them HH- or hyper-heavenly spaces; the name originates in the fact that they are a
`generalization' of the H-spaces, or heavens, introduced by
NewmanThis `generalization' is in the sense that while an H-space is a Ricci-flat,
complex 4-manifold with self-dual curvature, an HH-space was originally defined as a Ricci-flat, complex 4-manifold
with (half) algebraically special curvature.
But the definition of Newman's H-space is more subtle than this simple characterization in the sense that it is, by construction,
associated to general, asymptotically flat spacetimes (with real-analytic null infinity), via solutions to the good-cut equation.
See also section REF ..
Remarkably, Plebański and Robinson showed in [1]} that
the existence of such integral submanifolds is sufficient for reducing the full non-linear
Einstein vacuum equations to a scalar, second-order, non-linear PDE known as `hyper-heavenly equation'.
